Mathematical Logic
Mathematical Logic is a way of performing operations in Maths . It is much like functions , but the basic difference is that in functions , we perform an operation over the equation of the y coordinate , while in logic , we perform an operation over the inputs of a circuit . Logic Logic is an operation performed over the inputs of a circuit to give an output . There are many such different logical operators in Mathematics and digital electronics . The logic operators included in the syllabus are mentioned below . The number of possible outcomes is given by 2 raised to number of inputs . 'AND Logic' In AND logic , the output will be true only when all inputs are true . The output will be false even if one input is false . It is denoted by ^ . ex. p^q . Truth table 'OR Logic' In OR Logic , the output will be true even if one input is true . The output will be false only if all the inputs are false . It is denoted by v . ex. pvq 'NOT Logic' In NOT Logic , the output will be the opposite of input. If the input is true , the output will be false . NOT logic is also referred to as Compliment or negation . It is denoted by ~ . ex. ~p . Note : NOT logic has only one input . 'Implication' Under implication , the output will be false only if the first input is true and second input is false . in the rest cases , output will always be True . It is denoted by → . ex. p→q . Note : Implication has only two inputs . Truth Table The equation of implication is given as :- p→q≈~pvq TruthTable 'Double Implication' Under double implication , the output is true , only when the number of "True" inputs is even . If there are 0 or 2 or 4 .... True values , the output will be true . It is denoted by ←→ . ex. p ←→ q . Logic Circuits In logic circuits , OR stands for parallel and AND stands for series connections . Simplification p^T =p p^F =F pvT =T pvF =p p^~p=F pv~p=T ~(p^q) = ~pv~q ~(pvq) = ~p^~q Statements A statement is a declarative sentence which is either true or false . Assertive sentences are statement . Imperative , exclamatory and interrogative sentences are not statements . Logical Connectives Converse , Inverse and Contrapositive 'Converse : '''q ---> p '''Inverse : '~p ---> ~q 'Contrapositive : '~q ---> ~p Tautology , Contradiction and Contingency '''Tautology : '''When all outputs are true '''Contradiction : '''When all outputs are false '''Contingency : '''When the outputs are a combination of true and false values Quantifiers Universal Quantifier : ∀ Existential Quantifier : ∃ Principle of Duality Under the principle of duality , the logic signs are simply interchanged . AND becomes OR . OR becomes AND . Tautology becomes contradiction . Contradiction becomes Tautology . Negation of Compound Statements (De Morgan's Theorems ) ~(pvq) = ~p^~q ~(p^q) = ~pv~q Category:Mathematics